LMFDB - Embedded newform 8028.2.h.a.4013.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8028,2,Mod(4013,8028)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8028.4013");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8028 = 2^{2} \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8028.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$64.1039027427$
Analytic rank:	$0$
Dimension:	$76$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4013.6
Character	$\chi$	$=$	8028.4013
Dual form			8028.2.h.a.4013.5

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.83180 q^{5} -1.12103 q^{7} +O(q^{10})$	$q-3.83180 q^{5} -1.12103 q^{7} -3.85559 q^{11} -2.61838i q^{13} +4.98618i q^{17} -3.05155 q^{19} +6.94428 q^{23} +9.68269 q^{25} +9.74680i q^{29} +9.63763 q^{31} +4.29556 q^{35} +7.68305 q^{37} -9.19671i q^{41} -6.95420 q^{43} -3.77361i q^{47} -5.74329 q^{49} +1.59397i q^{53} +14.7738 q^{55} -1.99264 q^{59} -2.00814i q^{61} +10.0331i q^{65} -12.9329i q^{67} +10.0543 q^{71} +13.4710 q^{73} +4.32223 q^{77} -13.2103i q^{79} +3.05738i q^{83} -19.1060i q^{85} +7.93890i q^{89} +2.93529i q^{91} +11.6929 q^{95} -1.37660i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 76 q + 8 q^{7}+O(q^{10}) $ 76 * q + 8 * q^7	$ 76 q + 8 q^{7} + 16 q^{19} + 100 q^{25} - 8 q^{31} + 32 q^{37} - 24 q^{43} + 68 q^{49} + 24 q^{55} + 8 q^{73}+O(q^{100}) $ 76 * q + 8 * q^7 + 16 * q^19 + 100 * q^25 - 8 * q^31 + 32 * q^37 - 24 * q^43 + 68 * q^49 + 24 * q^55 + 8 * q^73

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8028\mathbb{Z}\right)^\times$.

$n$	$893$	$2233$	$4015$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.83180			−1.71363			−0.856816	−	0.515621i	$-0.827561\pi$
$5$	−3.83180			−1.71363			−0.856816	+	0.515621i	$0.827561\pi$
$6$		0			0
$7$	−1.12103			−0.423710			−0.211855	−	0.977301i	$-0.567950\pi$
$7$	−1.12103			−0.423710			−0.211855	+	0.977301i	$0.567950\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.85559			−1.16250			−0.581252	−	0.813724i	$-0.697437\pi$
$11$	−3.85559			−1.16250			−0.581252	+	0.813724i	$0.697437\pi$
$12$		0			0
$13$		−	2.61838i		−	0.726209i	−0.931748	−	0.363105i	$-0.881717\pi$
$13$		−	2.61838i		−	0.726209i	0.931748	−	0.363105i	$-0.118283\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.98618i			1.20933i	0.796482	+	0.604663i	$0.206692\pi$
$17$			4.98618i			1.20933i	−0.796482	+	0.604663i	$0.793308\pi$
$18$		0			0
$19$	−3.05155			−0.700074			−0.350037	−	0.936736i	$-0.613831\pi$
$19$	−3.05155			−0.700074			−0.350037	+	0.936736i	$0.613831\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.94428			1.44798			0.723991	−	0.689810i	$-0.242306\pi$
$23$	6.94428			1.44798			0.723991	+	0.689810i	$0.242306\pi$
$24$		0			0
$25$	9.68269			1.93654
$26$		0			0
$27$		0			0
$28$		0			0
$29$			9.74680i			1.80993i	0.425481	+	0.904967i	$0.360105\pi$
$29$			9.74680i			1.80993i	−0.425481	+	0.904967i	$0.639895\pi$
$30$		0			0
$31$	9.63763			1.73097			0.865484	−	0.500936i	$-0.167011\pi$
$31$	9.63763			1.73097			0.865484	+	0.500936i	$0.167011\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.29556			0.726083
$36$		0			0
$37$	7.68305			1.26309			0.631543	−	0.775341i	$-0.282422\pi$
$37$	7.68305			1.26309			0.631543	+	0.775341i	$0.282422\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	9.19671i		−	1.43628i	−0.695896	−	0.718142i	$-0.744993\pi$
$41$		−	9.19671i		−	1.43628i	0.695896	−	0.718142i	$-0.255007\pi$
$42$		0			0
$43$	−6.95420			−1.06051			−0.530253	−	0.847839i	$-0.677903\pi$
$43$	−6.95420			−1.06051			−0.530253	+	0.847839i	$0.677903\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	3.77361i		−	0.550438i	−0.961382	−	0.275219i	$-0.911250\pi$
$47$		−	3.77361i		−	0.550438i	0.961382	−	0.275219i	$-0.0887504\pi$
$48$		0			0
$49$	−5.74329			−0.820470
$50$		0			0
$51$		0			0
$52$		0			0
$53$			1.59397i			0.218949i	0.993990	+	0.109474i	$0.0349167\pi$
$53$			1.59397i			0.218949i	−0.993990	+	0.109474i	$0.965083\pi$
$54$		0			0
$55$	14.7738			1.99210
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.99264			−0.259420			−0.129710	−	0.991552i	$-0.541405\pi$
$59$	−1.99264			−0.259420			−0.129710	+	0.991552i	$0.541405\pi$
$60$		0			0
$61$		−	2.00814i		−	0.257116i	−0.991702	−	0.128558i	$-0.958965\pi$
$61$		−	2.00814i		−	0.257116i	0.991702	−	0.128558i	$-0.0410348\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			10.0331i			1.24446i
$66$		0			0
$67$		−	12.9329i		−	1.58000i	−0.613104	−	0.790002i	$-0.710079\pi$
$67$		−	12.9329i		−	1.58000i	0.613104	−	0.790002i	$-0.289921\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.0543			1.19323			0.596615	−	0.802528i	$-0.296512\pi$
$71$	10.0543			1.19323			0.596615	+	0.802528i	$0.296512\pi$
$72$		0			0
$73$	13.4710			1.57666			0.788329	−	0.615253i	$-0.210946\pi$
$73$	13.4710			1.57666			0.788329	+	0.615253i	$0.210946\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.32223			0.492564
$78$		0			0
$79$		−	13.2103i		−	1.48627i	−0.669139	−	0.743137i	$-0.733337\pi$
$79$		−	13.2103i		−	1.48627i	0.669139	−	0.743137i	$-0.266663\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.05738i			0.335591i	0.985822	+	0.167796i	$0.0536649\pi$
$83$			3.05738i			0.335591i	−0.985822	+	0.167796i	$0.946335\pi$
$84$		0			0
$85$		−	19.1060i		−	2.07234i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			7.93890i			0.841522i	0.907172	+	0.420761i	$0.138237\pi$
$89$			7.93890i			0.841522i	−0.907172	+	0.420761i	$0.861763\pi$
$90$		0			0
$91$			2.93529i			0.307702i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	11.6929			1.19967
$96$		0			0
$97$		−	1.37660i		−	0.139772i	−0.997555	−	0.0698861i	$-0.977736\pi$
$97$		−	1.37660i		−	0.139772i	0.997555	−	0.0698861i	$-0.0222636\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			6.62678i			0.659390i	0.944088	+	0.329695i	$0.106946\pi$
$101$			6.62678i			0.659390i	−0.944088	+	0.329695i	$0.893054\pi$
$102$		0			0
$103$			16.3867i			1.61463i	0.590121	+	0.807315i	$0.299080\pi$
$103$			16.3867i			1.61463i	−0.590121	+	0.807315i	$0.700920\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.59425			0.250796			0.125398	−	0.992107i	$-0.459979\pi$
$107$	2.59425			0.250796			0.125398	+	0.992107i	$0.459979\pi$
$108$		0			0
$109$	−18.8132			−1.80198			−0.900988	−	0.433843i	$-0.857157\pi$
$109$	−18.8132			−1.80198			−0.900988	+	0.433843i	$0.857157\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.18560			0.299676			0.149838	−	0.988711i	$-0.452125\pi$
$113$	3.18560			0.299676			0.149838	+	0.988711i	$0.452125\pi$
$114$		0			0
$115$	−26.6091			−2.48131
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	5.58966i		−	0.512403i
$120$		0			0
$121$	3.86557			0.351415
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−17.9431			−1.60488
$126$		0			0
$127$	−1.86418			−0.165420			−0.0827098	−	0.996574i	$-0.526357\pi$
$127$	−1.86418			−0.165420			−0.0827098	+	0.996574i	$0.526357\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			13.6733i			1.19464i	0.802004	+	0.597319i	$0.203767\pi$
$131$			13.6733i			1.19464i	−0.802004	+	0.597319i	$0.796233\pi$
$132$		0			0
$133$	3.42088			0.296628
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−13.0448			−1.11449			−0.557246	−	0.830348i	$-0.688142\pi$
$137$	−13.0448			−1.11449			−0.557246	+	0.830348i	$0.688142\pi$
$138$		0			0
$139$	−0.652879			−0.0553764			−0.0276882	−	0.999617i	$-0.508815\pi$
$139$	−0.652879			−0.0553764			−0.0276882	+	0.999617i	$0.508815\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			10.0954i			0.844221i
$144$		0			0
$145$		−	37.3478i		−	3.10156i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−18.1007			−1.48286			−0.741432	−	0.671028i	$-0.765853\pi$
$149$	−18.1007			−1.48286			−0.741432	+	0.671028i	$0.765853\pi$
$150$		0			0
$151$			17.7551i			1.44489i	0.691427	+	0.722447i	$0.256982\pi$
$151$			17.7551i			1.44489i	−0.691427	+	0.722447i	$0.743018\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−36.9295			−2.96625
$156$		0			0
$157$			2.17829i			0.173846i	0.996215	+	0.0869231i	$0.0277034\pi$
$157$			2.17829i			0.173846i	−0.996215	+	0.0869231i	$0.972297\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−7.78475			−0.613524
$162$		0			0
$163$			2.31000i			0.180933i	0.995900	+	0.0904664i	$0.0288358\pi$
$163$			2.31000i			0.180933i	−0.995900	+	0.0904664i	$0.971164\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.33801			−0.335685			−0.167843	−	0.985814i	$-0.553680\pi$
$167$	−4.33801			−0.335685			−0.167843	+	0.985814i	$0.553680\pi$
$168$		0			0
$169$	6.14407			0.472620
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.366657			−0.0278764			−0.0139382	−	0.999903i	$-0.504437\pi$
$173$	−0.366657			−0.0278764			−0.0139382	+	0.999903i	$0.504437\pi$
$174$		0			0
$175$	−10.8546			−0.820530
$176$		0			0
$177$		0			0
$178$		0			0
$179$			2.52399i			0.188652i	0.995541	+	0.0943260i	$0.0300696\pi$
$179$			2.52399i			0.188652i	−0.995541	+	0.0943260i	$0.969930\pi$
$180$		0			0
$181$	−17.1178			−1.27235			−0.636176	−	0.771544i	$-0.719485\pi$
$181$	−17.1178			−1.27235			−0.636176	+	0.771544i	$0.719485\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−29.4399			−2.16447
$186$		0			0
$187$		−	19.2246i		−	1.40585i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.9360			−0.791303			−0.395651	−	0.918401i	$-0.629481\pi$
$191$	−10.9360			−0.791303			−0.395651	+	0.918401i	$0.629481\pi$
$192$		0			0
$193$			22.2809i			1.60381i	0.597449	+	0.801907i	$0.296181\pi$
$193$			22.2809i			1.60381i	−0.597449	+	0.801907i	$0.703819\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			12.9644i			0.923678i	0.886964	+	0.461839i	$0.152810\pi$
$197$			12.9644i			0.923678i	−0.886964	+	0.461839i	$0.847190\pi$
$198$		0			0
$199$	−13.9552			−0.989261			−0.494630	−	0.869103i	$-0.664697\pi$
$199$	−13.9552			−0.989261			−0.494630	+	0.869103i	$0.664697\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	10.9265i		−	0.766887i
$204$		0			0
$205$			35.2400i			2.46127i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	11.7655			0.813839
$210$		0			0
$211$	6.03450			0.415432			0.207716	−	0.978189i	$-0.433397\pi$
$211$	6.03450			0.415432			0.207716	+	0.978189i	$0.433397\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	26.6471			1.81732
$216$		0			0
$217$	−10.8041			−0.733428
$218$		0			0
$219$		0			0
$220$		0			0
$221$	13.0557			0.878223
$222$		0			0
$223$	8.12949	−	12.5264i	0.544391	−	0.838832i
$223$	8.12949	−	12.5264i	0.544391	−	0.838832i
$224$		0			0
$225$		0			0
$226$		0			0
$227$			11.8556i			0.786882i	0.919350	+	0.393441i	$0.128715\pi$
$227$			11.8556i			0.786882i	−0.919350	+	0.393441i	$0.871285\pi$
$228$		0			0
$229$		−	15.8023i		−	1.04425i	−0.852870	−	0.522124i	$-0.825140\pi$
$229$		−	15.8023i		−	1.04425i	0.852870	−	0.522124i	$-0.174860\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.328116			−0.0214956			−0.0107478	−	0.999942i	$-0.503421\pi$
$233$	−0.328116			−0.0214956			−0.0107478	+	0.999942i	$0.503421\pi$
$234$		0			0
$235$			14.4597i			0.943249i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	16.6350i		−	1.07603i	−0.842935	−	0.538015i	$-0.819174\pi$
$239$		−	16.6350i		−	1.07603i	0.842935	−	0.538015i	$-0.180826\pi$
$240$		0			0
$241$	10.7983			0.695581			0.347790	−	0.937572i	$-0.386932\pi$
$241$	10.7983			0.695581			0.347790	+	0.937572i	$0.386932\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	22.0071			1.40598
$246$		0			0
$247$			7.99013i			0.508400i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	21.7073i		−	1.37015i	−0.728470	−	0.685077i	$-0.759768\pi$
$251$		−	21.7073i		−	1.37015i	0.728470	−	0.685077i	$-0.240232\pi$
$252$		0			0
$253$	−26.7743			−1.68328
$254$		0			0
$255$		0			0
$256$		0			0
$257$			20.8001i			1.29748i	0.761011	+	0.648738i	$0.224703\pi$
$257$			20.8001i			1.29748i	−0.761011	+	0.648738i	$0.775297\pi$
$258$		0			0
$259$	−8.61293			−0.535182
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−21.1353			−1.30326			−0.651628	−	0.758538i	$-0.725914\pi$
$263$	−21.1353			−1.30326			−0.651628	+	0.758538i	$0.725914\pi$
$264$		0			0
$265$		−	6.10777i		−	0.375197i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.4455			−0.636875			−0.318437	−	0.947944i	$-0.603158\pi$
$269$	−10.4455			−0.636875			−0.318437	+	0.947944i	$0.603158\pi$
$270$		0			0
$271$		−	2.00917i		−	0.122048i	−0.998136	−	0.0610242i	$-0.980563\pi$
$271$		−	2.00917i		−	0.122048i	0.998136	−	0.0610242i	$-0.0194367\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−37.3325			−2.25123
$276$		0			0
$277$		−	23.7401i		−	1.42640i	−0.700959	−	0.713202i	$-0.747244\pi$
$277$		−	23.7401i		−	1.42640i	0.700959	−	0.713202i	$-0.252756\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	31.8965i		−	1.90279i	−0.307980	−	0.951393i	$-0.599653\pi$
$281$		−	31.8965i		−	1.90279i	0.307980	−	0.951393i	$-0.400347\pi$
$282$		0			0
$283$	4.36152			0.259265			0.129633	−	0.991562i	$-0.458620\pi$
$283$	4.36152			0.259265			0.129633	+	0.991562i	$0.458620\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			10.3098i			0.608568i
$288$		0			0
$289$	−7.86195			−0.462468
$290$		0			0
$291$		0			0
$292$		0			0
$293$	23.1255			1.35101			0.675504	−	0.737356i	$-0.263926\pi$
$293$	23.1255			1.35101			0.675504	+	0.737356i	$0.263926\pi$
$294$		0			0
$295$	7.63541			0.444551
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	18.1828i		−	1.05154i
$300$		0			0
$301$	7.79587			0.449347
$302$		0			0
$303$		0			0
$304$		0			0
$305$			7.69478i			0.440602i
$306$		0			0
$307$			7.18911i			0.410304i	0.978730	+	0.205152i	$0.0657689\pi$
$307$			7.18911i			0.410304i	−0.978730	+	0.205152i	$0.934231\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.48967			0.0844713			0.0422356	−	0.999108i	$-0.486552\pi$
$311$	1.48967			0.0844713			0.0422356	+	0.999108i	$0.486552\pi$
$312$		0			0
$313$		−	1.42993i		−	0.0808245i	−0.999183	−	0.0404122i	$-0.987133\pi$
$313$		−	1.42993i		−	0.0808245i	0.999183	−	0.0404122i	$-0.0128671\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	15.6700i		−	0.880116i	−0.897969	−	0.440058i	$-0.854958\pi$
$317$		−	15.6700i		−	0.880116i	0.897969	−	0.440058i	$-0.145042\pi$
$318$		0			0
$319$		−	37.5796i		−	2.10406i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	15.2156i		−	0.846617i
$324$		0			0
$325$		−	25.3530i		−	1.40633i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			4.23034i			0.233226i
$330$		0			0
$331$			33.7059i			1.85264i	0.376735	+	0.926321i	$0.377047\pi$
$331$			33.7059i			1.85264i	−0.376735	+	0.926321i	$0.622953\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			49.5563i			2.70755i
$336$		0			0
$337$			4.65679i			0.253672i	0.991924	+	0.126836i	$0.0404821\pi$
$337$			4.65679i			0.253672i	−0.991924	+	0.126836i	$0.959518\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−37.1587			−2.01226
$342$		0			0
$343$	14.2856			0.771351
$344$		0			0
$345$		0			0
$346$		0			0
$347$			12.6543i			0.679316i	0.940549	+	0.339658i	$0.110311\pi$
$347$			12.6543i			0.679316i	−0.940549	+	0.339658i	$0.889689\pi$
$348$		0			0
$349$	−0.561956			−0.0300808			−0.0150404	−	0.999887i	$-0.504788\pi$
$349$	−0.561956			−0.0300808			−0.0150404	+	0.999887i	$0.504788\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	14.6361i		−	0.779001i	−0.921026	−	0.389500i	$-0.872648\pi$
$353$		−	14.6361i		−	0.779001i	0.921026	−	0.389500i	$-0.127352\pi$
$354$		0			0
$355$	−38.5262			−2.04476
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	17.8782i		−	0.943574i	−0.881713	−	0.471787i	$-0.843609\pi$
$359$		−	17.8782i		−	0.943574i	0.881713	−	0.471787i	$-0.156391\pi$
$360$		0			0
$361$	−9.68804			−0.509897
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−51.6181			−2.70181
$366$		0			0
$367$	−18.1446			−0.947140			−0.473570	−	0.880756i	$-0.657035\pi$
$367$	−18.1446			−0.947140			−0.473570	+	0.880756i	$0.657035\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	1.78689i		−	0.0927706i
$372$		0			0
$373$		−	10.7034i		−	0.554204i	−0.960841	−	0.277102i	$-0.910626\pi$
$373$		−	10.7034i		−	0.554204i	0.960841	−	0.277102i	$-0.0893739\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	25.5209			1.31439
$378$		0			0
$379$	8.17980			0.420168			0.210084	−	0.977683i	$-0.432626\pi$
$379$	8.17980			0.420168			0.210084	+	0.977683i	$0.432626\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−29.7035			−1.51778			−0.758888	−	0.651221i	$-0.774257\pi$
$383$	−29.7035			−1.51778			−0.758888	+	0.651221i	$0.774257\pi$
$384$		0			0
$385$	−16.5619			−0.844074
$386$		0			0
$387$		0			0
$388$		0			0
$389$			1.82829i			0.0926981i	0.998925	+	0.0463490i	$0.0147586\pi$
$389$			1.82829i			0.0926981i	−0.998925	+	0.0463490i	$0.985241\pi$
$390$		0			0
$391$			34.6254i			1.75108i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			50.6192i			2.54693i
$396$		0			0
$397$		−	32.6536i		−	1.63883i	−0.573197	−	0.819417i	$-0.694297\pi$
$397$		−	32.6536i		−	1.63883i	0.573197	−	0.819417i	$-0.305703\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			31.1978i			1.55795i	0.627058	+	0.778973i	$0.284259\pi$
$401$			31.1978i			1.55795i	−0.627058	+	0.778973i	$0.715741\pi$
$402$		0			0
$403$		−	25.2350i		−	1.25705i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−29.6227			−1.46834
$408$		0			0
$409$		−	5.98010i		−	0.295697i	−0.989010	−	0.147848i	$-0.952765\pi$
$409$		−	5.98010i		−	0.295697i	0.989010	−	0.147848i	$-0.0472348\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.23381			0.109919
$414$		0			0
$415$		−	11.7153i		−	0.575080i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			3.05350i			0.149173i	0.997215	+	0.0745865i	$0.0237637\pi$
$419$			3.05350i			0.149173i	−0.997215	+	0.0745865i	$0.976236\pi$
$420$		0			0
$421$			39.1719i			1.90912i	0.298015	+	0.954561i	$0.403675\pi$
$421$			39.1719i			1.90912i	−0.298015	+	0.954561i	$0.596325\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			48.2796i			2.34190i
$426$		0			0
$427$			2.25118i			0.108942i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−30.4950			−1.46889			−0.734447	−	0.678667i	$-0.762558\pi$
$431$	−30.4950			−1.46889			−0.734447	+	0.678667i	$0.762558\pi$
$432$		0			0
$433$	32.3631			1.55527			0.777636	−	0.628715i	$-0.216419\pi$
$433$	32.3631			1.55527			0.777636	+	0.628715i	$0.216419\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−21.1908			−1.01369
$438$		0			0
$439$		−	7.97903i		−	0.380818i	−0.981705	−	0.190409i	$-0.939019\pi$
$439$		−	7.97903i		−	0.380818i	0.981705	−	0.190409i	$-0.0609814\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	33.1779i		−	1.57633i	−0.615463	−	0.788166i	$-0.711031\pi$
$443$		−	33.1779i		−	1.57633i	0.615463	−	0.788166i	$-0.288969\pi$
$444$		0			0
$445$		−	30.4203i		−	1.44206i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−22.9806			−1.08452			−0.542260	−	0.840211i	$-0.682431\pi$
$449$	−22.9806			−1.08452			−0.542260	+	0.840211i	$0.682431\pi$
$450$		0			0
$451$			35.4587i			1.66969i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	11.2474i		−	0.527288i
$456$		0			0
$457$		−	32.9030i		−	1.53914i	−0.638565	−	0.769568i	$-0.720472\pi$
$457$		−	32.9030i		−	1.53914i	0.638565	−	0.769568i	$-0.279528\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			9.87361i			0.459860i	0.973207	+	0.229930i	$0.0738497\pi$
$461$			9.87361i			0.459860i	−0.973207	+	0.229930i	$0.926150\pi$
$462$		0			0
$463$	−33.3274			−1.54885			−0.774427	−	0.632664i	$-0.781962\pi$
$463$	−33.3274			−1.54885			−0.774427	+	0.632664i	$0.781962\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−11.5782			−0.535777			−0.267888	−	0.963450i	$-0.586326\pi$
$467$	−11.5782			−0.535777			−0.267888	+	0.963450i	$0.586326\pi$
$468$		0			0
$469$			14.4982i			0.669463i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	26.8126			1.23284
$474$		0			0
$475$	−29.5472			−1.35572
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	34.1506i		−	1.56038i	−0.625543	−	0.780190i	$-0.715123\pi$
$479$		−	34.1506i		−	1.56038i	0.625543	−	0.780190i	$-0.284877\pi$
$480$		0			0
$481$		−	20.1172i		−	0.917264i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			5.27484i			0.239518i
$486$		0			0
$487$	37.6562			1.70637			0.853183	−	0.521612i	$-0.174669\pi$
$487$	37.6562			1.70637			0.853183	+	0.521612i	$0.174669\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	22.8711			1.03216			0.516080	−	0.856541i	$-0.327391\pi$
$491$	22.8711			1.03216			0.516080	+	0.856541i	$0.327391\pi$
$492$		0			0
$493$	−48.5993			−2.18880
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−11.2712			−0.505583
$498$		0			0
$499$	2.24606			0.100547			0.0502737	−	0.998735i	$-0.483991\pi$
$499$	2.24606			0.100547			0.0502737	+	0.998735i	$0.483991\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.5406			0.559157			0.279578	−	0.960123i	$-0.409805\pi$
$503$	12.5406			0.559157			0.279578	+	0.960123i	$0.409805\pi$
$504$		0			0
$505$		−	25.3925i		−	1.12995i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			4.34476i			0.192578i	0.995353	+	0.0962891i	$0.0306973\pi$
$509$			4.34476i			0.192578i	−0.995353	+	0.0962891i	$0.969303\pi$
$510$		0			0
$511$	−15.1014			−0.668046
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	62.7906i		−	2.76688i
$516$		0			0
$517$			14.5495i			0.639886i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	16.5112			0.723370			0.361685	−	0.932300i	$-0.382202\pi$
$521$	16.5112			0.723370			0.361685	+	0.932300i	$0.382202\pi$
$522$		0			0
$523$		−	5.57482i		−	0.243770i	−0.992544	−	0.121885i	$-0.961106\pi$
$523$		−	5.57482i		−	0.243770i	0.992544	−	0.121885i	$-0.0388939\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			48.0549i			2.09330i
$528$		0			0
$529$	25.2230			1.09665
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−24.0805			−1.04304
$534$		0			0
$535$	−9.94066			−0.429772
$536$		0			0
$537$		0			0
$538$		0			0
$539$	22.1438			0.953800
$540$		0			0
$541$			18.0871i			0.777627i	0.921317	+	0.388813i	$0.127115\pi$
$541$			18.0871i			0.777627i	−0.921317	+	0.388813i	$0.872885\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	72.0884			3.08793
$546$		0			0
$547$	−21.4755			−0.918227			−0.459113	−	0.888378i	$-0.651833\pi$
$547$	−21.4755			−0.918227			−0.459113	+	0.888378i	$0.651833\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	29.7429i		−	1.26709i
$552$		0			0
$553$			14.8091i			0.629749i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	6.36597			0.269735			0.134867	−	0.990864i	$-0.456939\pi$
$557$	6.36597			0.269735			0.134867	+	0.990864i	$0.456939\pi$
$558$		0			0
$559$			18.2088i			0.770149i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7.48217			−0.315336			−0.157668	−	0.987492i	$-0.550398\pi$
$563$	−7.48217			−0.315336			−0.157668	+	0.987492i	$0.550398\pi$
$564$		0			0
$565$	−12.2066			−0.513535
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−43.9559			−1.84273			−0.921363	−	0.388703i	$-0.872923\pi$
$569$	−43.9559			−1.84273			−0.921363	+	0.388703i	$0.872923\pi$
$570$		0			0
$571$			6.95590i			0.291095i	0.989351	+	0.145548i	$0.0464944\pi$
$571$			6.95590i			0.291095i	−0.989351	+	0.145548i	$0.953506\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	67.2393			2.80407
$576$		0			0
$577$	3.71701			0.154741			0.0773706	−	0.997002i	$-0.475348\pi$
$577$	3.71701			0.154741			0.0773706	+	0.997002i	$0.475348\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	3.42742i		−	0.142193i
$582$		0			0
$583$		−	6.14569i		−	0.254528i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	8.50219			0.350923			0.175462	−	0.984486i	$-0.443858\pi$
$587$	8.50219			0.350923			0.175462	+	0.984486i	$0.443858\pi$
$588$		0			0
$589$	−29.4097			−1.21181
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−7.22316			−0.296619			−0.148310	−	0.988941i	$-0.547383\pi$
$593$	−7.22316			−0.296619			−0.148310	+	0.988941i	$0.547383\pi$
$594$		0			0
$595$			21.4184i			0.878071i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	10.5960i		−	0.432939i	−0.976289	−	0.216470i	$-0.930546\pi$
$599$		−	10.5960i		−	0.432939i	0.976289	−	0.216470i	$-0.0694542\pi$
$600$		0			0
$601$		−	46.0641i		−	1.87899i	−0.342556	−	0.939497i	$-0.611293\pi$
$601$		−	46.0641i		−	1.87899i	0.342556	−	0.939497i	$-0.388707\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−14.8121			−0.602197
$606$		0			0
$607$		−	28.6316i		−	1.16212i	−0.813861	−	0.581060i	$-0.802638\pi$
$607$		−	28.6316i		−	1.16212i	0.813861	−	0.581060i	$-0.197362\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.88077			−0.399733
$612$		0			0
$613$			19.8196i			0.800505i	0.916405	+	0.400252i	$0.131077\pi$
$613$			19.8196i			0.800505i	−0.916405	+	0.400252i	$0.868923\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	14.4003i		−	0.579734i	−0.957067	−	0.289867i	$-0.906389\pi$
$617$		−	14.4003i		−	0.579734i	0.957067	−	0.289867i	$-0.0936110\pi$
$618$		0			0
$619$		−	3.86311i		−	0.155271i	−0.996982	−	0.0776357i	$-0.975263\pi$
$619$		−	3.86311i		−	0.155271i	0.996982	−	0.0776357i	$-0.0247371\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	8.89975i		−	0.356561i
$624$		0			0
$625$	20.3410			0.813641
$626$		0			0
$627$		0			0
$628$		0			0
$629$			38.3090i			1.52748i
$630$		0			0
$631$			49.0085i			1.95100i	0.220009	+	0.975498i	$0.429391\pi$
$631$			49.0085i			1.95100i	−0.220009	+	0.975498i	$0.570609\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7.14318			0.283468
$636$		0			0
$637$			15.0381i			0.595833i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.72705			−0.265702			−0.132851	−	0.991136i	$-0.542413\pi$
$641$	−6.72705			−0.265702			−0.132851	+	0.991136i	$0.542413\pi$
$642$		0			0
$643$	21.2304			0.837243			0.418622	−	0.908161i	$-0.362513\pi$
$643$	21.2304			0.837243			0.418622	+	0.908161i	$0.362513\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			40.3811i			1.58754i	0.608215	+	0.793772i	$0.291886\pi$
$647$			40.3811i			1.58754i	−0.608215	+	0.793772i	$0.708114\pi$
$648$		0			0
$649$	7.68282			0.301577
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−44.4548			−1.73965			−0.869825	−	0.493360i	$-0.835769\pi$
$653$	−44.4548			−1.73965			−0.869825	+	0.493360i	$0.835769\pi$
$654$		0			0
$655$		−	52.3932i		−	2.04717i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	43.2453i		−	1.68460i	−0.539012	−	0.842298i	$-0.681202\pi$
$659$		−	43.2453i		−	1.68460i	0.539012	−	0.842298i	$-0.318798\pi$
$660$		0			0
$661$		−	28.4045i		−	1.10481i	−0.833577	−	0.552403i	$-0.813711\pi$
$661$		−	28.4045i		−	1.10481i	0.833577	−	0.552403i	$-0.186289\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−13.1081			−0.508312
$666$		0			0
$667$			67.6845i			2.62075i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			7.74256i			0.298898i
$672$		0			0
$673$	−32.4504			−1.25087			−0.625436	−	0.780275i	$-0.715079\pi$
$673$	−32.4504			−1.25087			−0.625436	+	0.780275i	$0.715079\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			12.4790i			0.479608i	0.970821	+	0.239804i	$0.0770832\pi$
$677$			12.4790i			0.479608i	−0.970821	+	0.239804i	$0.922917\pi$
$678$		0			0
$679$			1.54321i			0.0592228i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			39.0774i			1.49525i	0.664119	+	0.747627i	$0.268807\pi$
$683$			39.0774i			1.49525i	−0.664119	+	0.747627i	$0.731193\pi$
$684$		0			0
$685$	49.9850			1.90983
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.17362			0.159002
$690$		0			0
$691$			10.6246i			0.404178i	0.979367	+	0.202089i	$0.0647730\pi$
$691$			10.6246i			0.404178i	−0.979367	+	0.202089i	$0.935227\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.50170			0.0948949
$696$		0			0
$697$	45.8564			1.73694
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	12.8523i		−	0.485423i	−0.970099	−	0.242711i	$-0.921963\pi$
$701$		−	12.8523i		−	0.485423i	0.970099	−	0.242711i	$-0.0780368\pi$
$702$		0			0
$703$	−23.4452			−0.884253
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	7.42883i		−	0.279390i
$708$		0			0
$709$		−	30.4621i		−	1.14403i	−0.820244	−	0.572013i	$-0.806163\pi$
$709$		−	30.4621i		−	1.14403i	0.820244	−	0.572013i	$-0.193837\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	66.9263			2.50641
$714$		0			0
$715$		−	38.6836i		−	1.44668i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	25.3869i		−	0.946770i	−0.880856	−	0.473385i	$-0.843032\pi$
$719$		−	25.3869i		−	0.946770i	0.880856	−	0.473385i	$-0.156968\pi$
$720$		0			0
$721$		−	18.3700i		−	0.684135i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			94.3752i			3.50501i
$726$		0			0
$727$	−2.83731			−0.105230			−0.0526149	−	0.998615i	$-0.516756\pi$
$727$	−2.83731			−0.105230			−0.0526149	+	0.998615i	$0.516756\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	34.6749i		−	1.28250i
$732$		0			0
$733$	−17.0171			−0.628542			−0.314271	−	0.949333i	$-0.601760\pi$
$733$	−17.0171			−0.628542			−0.314271	+	0.949333i	$0.601760\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			49.8639i			1.83676i
$738$		0			0
$739$			25.8581i			0.951204i	0.879660	+	0.475602i	$0.157770\pi$
$739$			25.8581i			0.951204i	−0.879660	+	0.475602i	$0.842230\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			22.6817i			0.832113i	0.909339	+	0.416056i	$0.136588\pi$
$743$			22.6817i			0.832113i	−0.909339	+	0.416056i	$0.863412\pi$
$744$		0			0
$745$	69.3581			2.54108
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.90824			−0.106265
$750$		0			0
$751$	−12.8001			−0.467081			−0.233540	−	0.972347i	$-0.575031\pi$
$751$	−12.8001			−0.467081			−0.233540	+	0.972347i	$0.575031\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	68.0342i		−	2.47602i
$756$		0			0
$757$			45.6148i			1.65790i	0.559325	+	0.828948i	$0.311060\pi$
$757$			45.6148i			1.65790i	−0.559325	+	0.828948i	$0.688940\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−32.2279			−1.16826			−0.584129	−	0.811661i	$-0.698564\pi$
$761$	−32.2279			−1.16826			−0.584129	+	0.811661i	$0.698564\pi$
$762$		0			0
$763$	21.0902			0.763515
$764$		0			0
$765$		0			0
$766$		0			0
$767$			5.21751i			0.188393i
$768$		0			0
$769$	−16.7131			−0.602690			−0.301345	−	0.953515i	$-0.597436\pi$
$769$	−16.7131			−0.602690			−0.301345	+	0.953515i	$0.597436\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−24.6011			−0.884841			−0.442421	−	0.896808i	$-0.645880\pi$
$773$	−24.6011			−0.884841			−0.442421	+	0.896808i	$0.645880\pi$
$774$		0			0
$775$	93.3182			3.35209
$776$		0			0
$777$		0			0
$778$		0			0
$779$			28.0642i			1.00551i
$780$		0			0
$781$	−38.7654			−1.38713
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	8.34676i		−	0.297909i
$786$		0			0
$787$		−	36.0497i		−	1.28503i	−0.766272	−	0.642517i	$-0.777890\pi$
$787$		−	36.0497i		−	1.28503i	0.766272	−	0.642517i	$-0.222110\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.57116			−0.126976
$792$		0			0
$793$	−5.25808			−0.186720
$794$		0			0
$795$		0			0
$796$		0			0
$797$			25.2771i			0.895361i	0.894194	+	0.447681i	$0.147750\pi$
$797$			25.2771i			0.895361i	−0.894194	+	0.447681i	$0.852250\pi$
$798$		0			0
$799$	18.8159			0.665659
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−51.9386			−1.83287
$804$		0			0
$805$	29.8296			1.05135
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.86829			0.0656855			0.0328427	−	0.999461i	$-0.489544\pi$
$809$	1.86829			0.0656855			0.0328427	+	0.999461i	$0.489544\pi$
$810$		0			0
$811$		−	13.8514i		−	0.486387i	−0.969978	−	0.243194i	$-0.921805\pi$
$811$		−	13.8514i		−	0.486387i	0.969978	−	0.243194i	$-0.0781951\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	8.85144i		−	0.310053i
$816$		0			0
$817$	21.2211			0.742433
$818$		0			0
$819$		0			0
$820$		0			0
$821$			16.4106i			0.572734i	0.958120	+	0.286367i	$0.0924476\pi$
$821$			16.4106i			0.572734i	−0.958120	+	0.286367i	$0.907552\pi$
$822$		0			0
$823$		−	19.1450i		−	0.667352i	−0.942688	−	0.333676i	$-0.891711\pi$
$823$		−	19.1450i		−	0.667352i	0.942688	−	0.333676i	$-0.108289\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	49.3912			1.71750			0.858750	−	0.512395i	$-0.171242\pi$
$827$	49.3912			1.71750			0.858750	+	0.512395i	$0.171242\pi$
$828$		0			0
$829$			41.1731i			1.43000i	0.699124	+	0.715001i	$0.253574\pi$
$829$			41.1731i			1.43000i	−0.699124	+	0.715001i	$0.746426\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	28.6371i		−	0.992215i
$834$		0			0
$835$	16.6224			0.575241
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−44.4324			−1.53398			−0.766989	−	0.641660i	$-0.778246\pi$
$839$	−44.4324			−1.53398			−0.766989	+	0.641660i	$0.778246\pi$
$840$		0			0
$841$	−66.0001			−2.27586
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−23.5428			−0.809898
$846$		0			0
$847$	−4.33342			−0.148898
$848$		0			0
$849$		0			0
$850$		0			0
$851$	53.3532			1.82892
$852$		0			0
$853$			0.638814i			0.0218726i	0.999940	+	0.0109363i	$0.00348120\pi$
$853$			0.638814i			0.0218726i	−0.999940	+	0.0109363i	$0.996519\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			31.6326i			1.08055i	0.841489	+	0.540275i	$0.181680\pi$
$857$			31.6326i			1.08055i	−0.841489	+	0.540275i	$0.818320\pi$
$858$		0			0
$859$			16.0634i			0.548076i	0.961719	+	0.274038i	$0.0883595\pi$
$859$			16.0634i			0.548076i	−0.961719	+	0.274038i	$0.911641\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10.9022			−0.371117			−0.185558	−	0.982633i	$-0.559409\pi$
$863$	−10.9022			−0.371117			−0.185558	+	0.982633i	$0.559409\pi$
$864$		0			0
$865$	1.40496			0.0477699
$866$		0			0
$867$		0			0
$868$		0			0
$869$			50.9335i			1.72780i
$870$		0			0
$871$	−33.8633			−1.14741
$872$		0			0
$873$		0			0
$874$		0			0
$875$	20.1148			0.680004
$876$		0			0
$877$		−	2.37305i		−	0.0801322i	−0.999197	−	0.0400661i	$-0.987243\pi$
$877$		−	2.37305i		−	0.0801322i	0.999197	−	0.0400661i	$-0.0127569\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	49.9104i		−	1.68152i	−0.541405	−	0.840762i	$-0.682108\pi$
$881$		−	49.9104i		−	1.68152i	0.541405	−	0.840762i	$-0.317892\pi$
$882$		0			0
$883$		−	1.67907i		−	0.0565051i	−0.999601	−	0.0282525i	$-0.991006\pi$
$883$		−	1.67907i		−	0.0565051i	0.999601	−	0.0282525i	$-0.00899426\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	12.4887i		−	0.419328i	−0.977773	−	0.209664i	$-0.932763\pi$
$887$		−	12.4887i		−	0.419328i	0.977773	−	0.209664i	$-0.0672371\pi$
$888$		0			0
$889$	2.08981			0.0700899
$890$		0			0
$891$		0			0
$892$		0			0
$893$			11.5154i			0.385347i
$894$		0			0
$895$		−	9.67143i		−	0.323280i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			93.9360i			3.13294i
$900$		0			0
$901$	−7.94781			−0.264780
$902$		0			0
$903$		0			0
$904$		0			0
$905$	65.5918			2.18035
$906$		0			0
$907$	19.9199			0.661430			0.330715	−	0.943731i	$-0.392710\pi$
$907$	19.9199			0.661430			0.330715	+	0.943731i	$0.392710\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	6.23178i		−	0.206468i	−0.994657	−	0.103234i	$-0.967081\pi$
$911$		−	6.23178i		−	0.206468i	0.994657	−	0.103234i	$-0.0329191\pi$
$912$		0			0
$913$		−	11.7880i		−	0.390126i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	15.3281i		−	0.506180i
$918$		0			0
$919$		−	13.3916i		−	0.441748i	−0.975302	−	0.220874i	$-0.929109\pi$
$919$		−	13.3916i		−	0.441748i	0.975302	−	0.220874i	$-0.0708910\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	26.3261i		−	0.866534i
$924$		0			0
$925$	74.3926			2.44601
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	9.97517i		−	0.327275i	−0.986521	−	0.163637i	$-0.947677\pi$
$929$		−	9.97517i		−	0.327275i	0.986521	−	0.163637i	$-0.0523227\pi$
$930$		0			0
$931$	17.5259			0.574390
$932$		0			0
$933$		0			0
$934$		0			0
$935$			73.6650i			2.40910i
$936$		0			0
$937$		−	5.68472i		−	0.185712i	−0.995680	−	0.0928559i	$-0.970400\pi$
$937$		−	5.68472i		−	0.185712i	0.995680	−	0.0928559i	$-0.0295996\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	19.5750i		−	0.638128i	−0.947733	−	0.319064i	$-0.896632\pi$
$941$		−	19.5750i		−	0.638128i	0.947733	−	0.319064i	$-0.103368\pi$
$942$		0			0
$943$		−	63.8645i		−	2.07971i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	36.1139i		−	1.17354i	−0.809753	−	0.586771i	$-0.800399\pi$
$947$		−	36.1139i		−	1.17354i	0.809753	−	0.586771i	$-0.199601\pi$
$948$		0			0
$949$		−	35.2722i		−	1.14498i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−40.3821			−1.30811			−0.654053	−	0.756449i	$-0.726933\pi$
$953$	−40.3821			−1.30811			−0.654053	+	0.756449i	$0.726933\pi$
$954$		0			0
$955$	41.9047			1.35600
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.6236			0.472221
$960$		0			0
$961$	61.8839			1.99625
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	85.3760i		−	2.74835i
$966$		0			0
$967$			0.630802i			0.0202852i	0.999949	+	0.0101426i	$0.00322855\pi$
$967$			0.630802i			0.0202852i	−0.999949	+	0.0101426i	$0.996771\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	41.5104			1.33213			0.666066	−	0.745892i	$-0.267977\pi$
$971$	41.5104			1.33213			0.666066	+	0.745892i	$0.267977\pi$
$972$		0			0
$973$	0.731897			0.0234635
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.87069			−0.123834			−0.0619171	−	0.998081i	$-0.519721\pi$
$977$	−3.87069			−0.123834			−0.0619171	+	0.998081i	$0.519721\pi$
$978$		0			0
$979$		−	30.6091i		−	0.978272i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	29.9844			0.956353			0.478177	−	0.878264i	$-0.341298\pi$
$983$	29.9844			0.956353			0.478177	+	0.878264i	$0.341298\pi$
$984$		0			0
$985$		−	49.6771i		−	1.58284i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−48.2919			−1.53559
$990$		0			0
$991$			2.64814i			0.0841208i	0.999115	+	0.0420604i	$0.0133922\pi$
$991$			2.64814i			0.0841208i	−0.999115	+	0.0420604i	$0.986608\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	53.4737			1.69523
$996$		0			0
$997$	−11.1249			−0.352328			−0.176164	−	0.984361i	$-0.556369\pi$
$997$	−11.1249			−0.352328			−0.176164	+	0.984361i	$0.556369\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8028.2.h.a.4013.6	yes	76
3.2	odd	2	inner	8028.2.h.a.4013.71	yes	76
223.222	odd	2	inner	8028.2.h.a.4013.72	yes	76
669.668	even	2	inner	8028.2.h.a.4013.5	✓	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8028.2.h.a.4013.5	✓	76	669.668	even	2	inner
8028.2.h.a.4013.6	yes	76	1.1	even	1	trivial
8028.2.h.a.4013.71	yes	76	3.2	odd	2	inner
8028.2.h.a.4013.72	yes	76	223.222	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8028.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.83180 q^{5} -1.12103 q^{7} +O(q^{10})\)	\(q-3.83180 q^{5} -1.12103 q^{7} -3.85559 q^{11} -2.61838i q^{13} +4.98618i q^{17} -3.05155 q^{19} +6.94428 q^{23} +9.68269 q^{25} +9.74680i q^{29} +9.63763 q^{31} +4.29556 q^{35} +7.68305 q^{37} -9.19671i q^{41} -6.95420 q^{43} -3.77361i q^{47} -5.74329 q^{49} +1.59397i q^{53} +14.7738 q^{55} -1.99264 q^{59} -2.00814i q^{61} +10.0331i q^{65} -12.9329i q^{67} +10.0543 q^{71} +13.4710 q^{73} +4.32223 q^{77} -13.2103i q^{79} +3.05738i q^{83} -19.1060i q^{85} +7.93890i q^{89} +2.93529i q^{91} +11.6929 q^{95} -1.37660i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q + 8 q^{7}+O(q^{10}) \) 76 * q + 8 * q^7	\( 76 q + 8 q^{7} + 16 q^{19} + 100 q^{25} - 8 q^{31} + 32 q^{37} - 24 q^{43} + 68 q^{49} + 24 q^{55} + 8 q^{73}+O(q^{100}) \) 76 * q + 8 * q^7 + 16 * q^19 + 100 * q^25 - 8 * q^31 + 32 * q^37 - 24 * q^43 + 68 * q^49 + 24 * q^55 + 8 * q^73

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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